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Second-Order Boundary Value Problem with Integral Boundary Conditions
Boundary Value Problems volume 2011, Article number: 260309 (2011)
Abstract
The nonlinear alternative of the Leray Schauder type and the Banach contraction principle are used to investigate the existence of solutions for second-order differential equations with integral boundary conditions. The compactness of solutions set is also investigated.
1. Introduction
This paper is concerned with the existence of solutions for the second-order boundary value problem

where is a given function and
is an integrable function.
Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [1–9] and the references therein. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example [10–14]. The goal of this paper is to give existence and uniqueness results for the problem (1.1). Our approach here is based on the Banach contraction principle and the Leray-Schauder alternative [15].
2. Preliminaries
In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. Let be the space of differentiable functions
whose first derivative,
, is absolutely continuous.
We take to be the Banach space of all continuous functions from
into
with the norm

and we let denote the Banach space of functions
that are Lebesgue integrable with norm

Definition 2.1.
A map is said to be
-Carathéodory if
(i) is measurable for each
(ii) is continuous for almost each
(iii)for every there exists
such that

3. Existence and Uniqueness Results
Definition 3.1.
A function is said to be a solution of (1.1) if
satisfies (1.1).
In what follows one assumes that One needs the following auxiliary result.
Lemma 3.2.
. Let . Then the function defined by

is the unique solution of the boundary value problem

where

Proof.
Let be a solution of the problem (3.2). Then integratingly, we obtain

Hence


where

Now, multiply (3.6) by and integrate over
, to get

Thus,

Substituting in (3.6) we have

Therefore

Set Note that

Our first result reads
Theorem 3.3.
Assume that is an
-Carathéodory function and the following hypothesis
(A1) There exists such that

holds. If

then the BVP (1.1) has a unique solution.
Proof.
Transform problem (1.1) into a fixed-point problem. Consider the operator defined by

We will show that is a contraction. Indeed, consider
Then we have for each

Therefore

showing that, is a contraction and hence it has a unique fixed point which is a solution to (1.1). The proof is completed.
We now present an existence result for problem (1.1).
Theorem 3.4.
Suppose that hypotheses
(H1) The function is an
-Carathéodory,
(H2) There exist functions and
such that

are satisfied. Then the BVP (1.1) has at least one solution. Moreover the solution set

is compact.
Proof.
Transform the BVP (1.1) into a fixed-point problem. Consider the operator as defined in Theorem 3.3. We will show that
satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.
Step 1 ( is continuous).
Let be a sequence such that
in
Then

Since is
-Carathéodory and
then

Hence

Step 2 ( maps bounded sets into bounded sets in
).
Indeed, it is enough to show that there exists a positive constant such that for each
one has
.
Let . Then for each
, we have

By (H2) we have for each

Then for each we have

Step 3 ( maps bounded set into equicontinuous sets of
).
Let ,
and
be a bounded set of
as in Step 2. Let
and
we have

As the right-hand side of the above inequality tends to zero. Then
is equicontinuous. As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem we can conclude that
is completely continuous.
Step 4 (A priori bounds on solutions).
Let for some
. This implies by
that for each
we have

Then

If we have

Thus

Hence

Set

and consider the operator From the choice of
, there is no
such that
for some
As a consequence of the nonlinear alternative of Leray-Schauder type [15], we deduce that
has a fixed point
in
which is a solution of the problem (1.1).
Now, prove that is compact. Let
be a sequence in
, then

As in Steps 3 and 4 we can easily prove that there exists such that

and the set is equicontinuous in
hence by Arzela-Ascoli theorem we can conclude that there exists a subsequence of
converging to
in
Using that fast that
is an
-Carathédory we can prove that

Thus is compact.
4. Examples
We present some examples to illustrate the applicability of our results.
Example 4.1.
Consider the following BVP

Set

We can easily show that conditions (A1), (3.14) are satisfied with

Hence, by Theorem 3.3, the BVP (4.1) has a unique solution on .
Example 4.2.
Consider the following BVP

Set

We can easily show that conditions (H1), (H2) are satisfied with

Hence, by Theorem 3.4, the BVP (4.4) has at least one solution on . Moreover, its solutions set is compact.
References
Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.
Belarbi A, Benchohra M: Existence results for nonlinear boundary-value problems with integral boundary conditions. Electronic Journal of Differential Equations 2005, 2005(06):10 .
Belarbi A, Benchohra M, Ouahab A: Multiple positive solutions for nonlinear boundary value problems with integral boundary conditions. Archivum Mathematicum 2008, 44(1):1-7.
Benchohra M, Hamani S, Nieto JJ: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions. The Rocky Mountain Journal of Mathematics 2010, 40(1):13-26. 10.1216/RMJ-2010-40-1-13
Infante G: Nonlocal boundary value problems with two nonlinear boundary conditions. Communications in Applied Analysis 2008, 12(3):279-288.
Lomtatidze A, Malaguti L: On a nonlocal boundary value problem for second order nonlinear singular differential equations. Georgian Mathematical Journal 2000, 7(1):133-154.
Webb JRL: Positive solutions of some higher order nonlocal boundary value problems. Electronic Journal of Qualitative Theory of Differential Equations 2009, (29):-15.
Webb JRL: A unified approach to nonlocal boundary value problems. In Dynamic Systems and Applications. Vol. 5. Dynamic, Atlanta, Ga, USA; 2008:510-515.
Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society 2006, 74(3):673-693. 10.1112/S0024610706023179
Brykalov SA: A second order nonlinear problem with two-point and integral boundary conditions. Georgian Mathematical Journal 1994, 1: 243-249. 10.1007/BF02254673
Denche M, Marhoune AL: High order mixed-type differential equations with weighted integral boundary conditions. Electronic Journal of Differential Equations 2000, 2000(60):-10.
Kiguradze I: Boundary value problems for systems of ordinary differential equations. Journal of Soviet Mathematics 1988, 43(2):2259-2339. 10.1007/BF01100360
Krall AM: The adjoint of a differential operator with integral boundary conditions. Proceedings of the American Mathematical Society 1965, 16: 738-742. 10.1090/S0002-9939-1965-0181794-9
Ma R: A survey on nonlocal boundary value problems. Applied Mathematics E-Notes 2007, 7: 257-279.
Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2003:xvi+690.
Acknowledgment
The authors are grateful to the referees for their remarks.
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Benchohra, M., Nieto, J. & Ouahab, A. Second-Order Boundary Value Problem with Integral Boundary Conditions. Bound Value Probl 2011, 260309 (2011). https://doi.org/10.1155/2011/260309
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DOI: https://doi.org/10.1155/2011/260309
Keywords
- Differential Equation
- Banach Space
- Unique Solution
- Ordinary Differential Equation
- Functional Equation